CLATPS(3)

okay so bestie, welcome to the CLATPS(3) page from the Library Functions Manual — it's the LAPACK subroutine that solves triangular systems when the matrix is packed, and it's literally out here preventing overflow like a main character. 💅✨

💖 NAME 💖

The source file this page is about is:

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/clatps.f

Yes, the whole NAME section is just that path. No cap. 🫶

✨ SYNOPSIS ✨

🛠️ Functions/Subroutines 🛠️

subroutine CLATPS (uplo, trans, diag, normin, n, ap, x, scale, cnorm, info)

CLATPS solves a triangular system of equations with the matrix held in packed storage. periodt. 💅

🎀 FUNCTION/SUBROUTINE DOCUMENTATION 🎀

subroutine CLATPS (character uplo, character trans, character diag, character normin, integer n, complex, dimension( * ) ap, complex, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)

CLATPS solves a triangular system of equations with the matrix held in packed storage.

🎯 Purpose 🎯

CLATPS solves one of the triangular systems

A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,

with scaling to prevent overflow, where A is an upper or lower triangular matrix stored in packed form. Here A**T denotes the transpose of A, A**H denotes the conjugate transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine CTPSV is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

🌸 Parameters 🌸

• UPLO — CHARACTER*1. Specifies whether the matrix A is upper or lower triangular. = 'U': Upper triangular; = 'L': Lower triangular.
• TRANS — CHARACTER*1. Specifies the operation applied to A. = 'N': Solve A * x = s*b (No transpose); = 'T': Solve A**T * x = s*b (Transpose); = 'C': Solve A**H * x = s*b (Conjugate transpose).
• DIAG — CHARACTER*1. Specifies whether or not the matrix A is unit triangular. = 'N': Non-unit triangular; = 'U': Unit triangular.
• NORMIN — CHARACTER*1. Specifies whether CNORM has been set or not. = 'Y': CNORM contains the column norms on entry; = 'N': CNORM is not set on entry. On exit, the norms will be computed and stored in CNORM.
• N — INTEGER. The order of the matrix A. N >= 0.
• AP — COMPLEX array, dimension (N*(N+1)/2). The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows:

if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

• X — COMPLEX array, dimension (N). On entry, the right hand side b of the triangular system. On exit, X is overwritten by the solution vector x.
• SCALE — REAL. The scaling factor s for the triangular system

A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.

If SCALE = 0, the matrix A is singular or badly scaled, and the vector x is an exact or approximate solution to A*x = 0.

• CNORM — REAL array, dimension (N). If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the off-diagonal part of the j-th column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1-norm of the offdiagonal part of the j-th column of A.
• INFO — INTEGER. = 0: successful exit; < 0: if INFO = -k, the k-th argument had an illegal value.

👑 Author 👑

The academic icons behind this:

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

🧚 Further Details 🧚

okay so here's the deeper lore, bestie — it's giving careful bound-checking energy:

A rough bound on x is computed; if that is less than overflow, CTPSV is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation.

A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is

x[1:n] := b[1:n]
for j = 1, ..., n
     x(j) := x(j) / A(j,j)
     x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
end

Define bounds on the components of x after j iterations of the loop:

M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]

Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

Then for iteration j+1 we have

M(j+1) <= G(j) / | A(j+1,j+1) |
G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
       <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

where CNORM(j+1) is greater than or equal to the infinity-norm of column j+1 of A, not counting the diagonal. Hence

G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
             1<=i<=j

and

|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                              1<=i< j

Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTPSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow).

The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

Similarly, a row-wise scheme is used to solve A**T *x = b or A**H *x = b. The basic algorithm for A upper triangular is

for j = 1, ..., n
     x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end

We simultaneously compute two bounds

G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j

The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is

M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

     <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
               1<=i<=j

and we can safely call CTPSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

📍 Definition 📍

Definition at line 229 of file clatps.f.

🪄 Author (auto-generated) 🪄

Generated automatically by Doxygen for LAPACK from the source code.

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💖 that's the whole page, bestie 💖 — LAPACK Version 3.12.0, from the Library Functions Manual, footer path /home/abui...RC/clatps.f(3). now go solve those triangular systems with your bad self. mwah 💋✨